#415513

order	512
nilpotency class	3
isoclinism	isoclinic class of #402446
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

688 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,5)(3,6,7,8)(10,16)(11,20)(12,24)(13,14)(17,30)(18,33)(19,27)(21,35)(25,26)(28,32)(34,36)(38,39);(2,5)(6,8)(9,10)(11,17,20,32)(12,18)(13,27)(14,16)(15,19)(21,31,34,40)(22,30,23,28)(24,29)(25,37)(26,33)(35,39,36,38);(1,3,4,7)(2,6,5,8)(9,11,13,22,15,20,14,23)(10,17,16,28,19,32,27,30)(12,21,25,35,26,34,24,36)(18,31,29,38,33,40,37,39);(9,12,14,24,15,26,13,25)(10,18,16,29,19,33,27,37)(11,21,23,36,20,34,22,35)(17,31,28,38,32,40,30,39)

Polycyclic presentation:

2^9|a^2=efgh,b^2=f,c^2=e,d^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322709);